Well, to start off, we have to figure out how to define mass. This is a lot harder than it sounds; I will provide three different definitions--in fact, in two of the three definitions, light is NOT massless!
If you read a textbook on classical physics or chemistry, or biology in general, there will be no reference to these definitions, and for a very good reason. Classical physics and chemistry, and biology, which is based on the principles of classical chemistry (I say classical chemistry as opposed to quantum chemistry), ignore the effects of relativity and quantum mechanics (besides the basic concepts like orbitals). This ends up making all three of these definitions the same, and in that context, mass it a property of mater. But, if you want to get more specific, here you go:
Definition 1: This definition is called "rest mass." This is the amount of inertia an object has (its resistance to acceleration) when it is not moving. This is a property of matter***(see below).
Definition 2: I'll call this "inertia," but many would call it "relativistic mass." You know what inertia is, I assume. Well, now that we know that the universe has a speed limit, the speed of light, how should we go about enforcing this speed limit? What happens at the speed of light that stops objects from going faster than it? At the speed of light, as you would guess, it is impossible to accelerate an object. Thus, has infinite inertia. In fact, an object gets harder to accelerate as it moves close to the speed of light. We can define an expression for mass that reflects the inertia of a moving object: m=m0/sqrt(1-v2/c2 ) where m0 is the rest mass. You can see that at low speeds, the intertia/relativistic mass is pretty much the rest mass. Now, have you tried speeding up or slowing down light? Pretty hard, huh? Light happens, in a way, to have an infinite amount of inertia; even though its rest mass is zero, the relativistic mass equation gives you a weird 0/0. So, that equation doesn't work that well for light. But, for other objects, because it accounts for the energy gained from motion (kinetic energy), energy is proportional to mass (it is not proportional to rest mass, as then you're ignoring kinetic energy).
Definition 3: Something that causes a gravitational pull. This isn't a great definition of mass, but one that may have crossed your mind. It turns out, thanks to General Relativity, all energy has a gravitational pull--including light. In that sense, one could maybe, maybe say that light has mass (I wouldn't agree with that statement, but I imagine one could argue semantics over it).
Now, back to definition 1. We know that an object has rest energy proportional to its rest mass--so is mass just energy or is it really a property of mater? Let's look deeper. A simple object's "rest mass" is actually mostly relativistic mass! Atoms are made of protons, neutrons, and electrons. Electrons get their mass mostly from the Higgs interaction (which gives the electron energy-so that mass is really a form of energy). Quarks, which make up protons and neutrons, get a small amount f mass from the Higgs interaction (I believe), but most of it comes from high-energy interactions with gluons. So, when you break it down, all mass, even the mass that we consider rest mass, comes from relativistic, high-energy interactions.
Given the nuances, I think its best that introductory textbooks skip the quark-gluon interactions and the Higgs interaction, and for low speeds just consider the mass of an object to be its rest mass, and call that a property of mater. This is, after all, how we perceive it, and classical physics is an approximation of the laws of physics for everyday situations.